Introducing Quintessence of Quadratics:
We began this project by reviewing topics like kinematics, projectile motion, pythagorean theorem, volume, and area. We took our knowledge we had about these topics and applied them to learn about quadratic equations and economics. The main problem was the rocket problem called "Victory Celebration". This problem was about a rocket that was going to be launched on an object 160 feet in the air. The questions we answered at the end of the project were: "what is the maximum height of the rocket?", "when does the rocket reach it's maximum height?", and"how long is the rocket in the air?". To answer these questions we started by looking at the equation: h(t)=d0+v0·t+(1/2) a·t^2. d0 (the initial distance), v0 (initial velocity, and a is the acceleration). We plugged in what we already had like the acceleration, initial velocity, and the starting height. To be able to answer the three questions, we completed worksheets that helped us with the skills we needed to solve them, We learned how quadratic equations help us find important parts of the problem like the x and y intercepts. The different quadratic equations we practiced were: vertex form, factored form, and standard form. We used the website called Desmos which helped us visualize the parabolas of the equations. This website helped us visualize the effects the equations has on the location, direction and size of the parabola. The objective was for us to develop our skills in solving quadratic equations, and to work with algebraic symbols, recognizing the connection between geometry and algebra.
Vertex form of the Quadratic Equation:
We started this section off by using the equation y=x^2, on the website Desmos. We figured out how adding a to the equation y=x^2 gives us y=ax^2. Adding different values can change the whole problem. For a, the value changes the concavity of the parabola. If it is higher you create a narrower curve, same goes for a lower value except the curve would be wider. A can't be zero because then the parabola would have no arch. If a is a negative number the parabola arches downwards in the -y axis.
Now moving on to h and k. H is equal to the x coordinate of the vertex and K is equal to the y coordinate of the vertex. By increasing or decreasing h's value, we shift the whole graph. If we increase or decrease k's value, we affect the y value which moves the vertex up or down. When we figured out k's purpose, we added it into the equation, Y=Ax^2+K. After we learned about h, we added that into the equation as well, making it a quadratic equation in vertex form: Y=A(x-H)^2+K.
Here are some examples of what I just explained:
We began this project by reviewing topics like kinematics, projectile motion, pythagorean theorem, volume, and area. We took our knowledge we had about these topics and applied them to learn about quadratic equations and economics. The main problem was the rocket problem called "Victory Celebration". This problem was about a rocket that was going to be launched on an object 160 feet in the air. The questions we answered at the end of the project were: "what is the maximum height of the rocket?", "when does the rocket reach it's maximum height?", and"how long is the rocket in the air?". To answer these questions we started by looking at the equation: h(t)=d0+v0·t+(1/2) a·t^2. d0 (the initial distance), v0 (initial velocity, and a is the acceleration). We plugged in what we already had like the acceleration, initial velocity, and the starting height. To be able to answer the three questions, we completed worksheets that helped us with the skills we needed to solve them, We learned how quadratic equations help us find important parts of the problem like the x and y intercepts. The different quadratic equations we practiced were: vertex form, factored form, and standard form. We used the website called Desmos which helped us visualize the parabolas of the equations. This website helped us visualize the effects the equations has on the location, direction and size of the parabola. The objective was for us to develop our skills in solving quadratic equations, and to work with algebraic symbols, recognizing the connection between geometry and algebra.
Vertex form of the Quadratic Equation:
We started this section off by using the equation y=x^2, on the website Desmos. We figured out how adding a to the equation y=x^2 gives us y=ax^2. Adding different values can change the whole problem. For a, the value changes the concavity of the parabola. If it is higher you create a narrower curve, same goes for a lower value except the curve would be wider. A can't be zero because then the parabola would have no arch. If a is a negative number the parabola arches downwards in the -y axis.
Now moving on to h and k. H is equal to the x coordinate of the vertex and K is equal to the y coordinate of the vertex. By increasing or decreasing h's value, we shift the whole graph. If we increase or decrease k's value, we affect the y value which moves the vertex up or down. When we figured out k's purpose, we added it into the equation, Y=Ax^2+K. After we learned about h, we added that into the equation as well, making it a quadratic equation in vertex form: Y=A(x-H)^2+K.
Here are some examples of what I just explained:
II. Exploring the Form of the Quadratic Equation:
After learning vertex form, we were then introduced to standard form and factored form. In standard form, the a, b, and c values are acting as a, k and h. Y=Ax^2+Bx+C is the quadratic equation in standard form. This equation gives us the y intercept value c. Factored form is y=a(x-p)(x-q). The p and q values are the x intercept values. We are sure they are x intercepts because in an x intercept the y value is 0, which makes (x-p)=0 or (x-q)=0. That means the intercepts are x=p and x=q. You are able to find the vertex from the factored form if you find the center point in between the x intercepts then you use that value to find the y. |
Converting Forms:
For Vertex to standard you have to take the (x+3)^2 and do (x+3) two times next to each other because the ^2 tells us so. You bring down the -2, and , you multiply 3 times 3 and get 9. You combine the like terms to get (x^2 +6x+9)-2. You are left with two x's which you can write down as y=x^2+6x+7
Standard to vertex: To turn an equation from standard to vertex you take a out and group the ax^2 and bx together. By using the area diagram you fill out the missing numbers and subtract the added term. You take the subtracted term and multiply by a. Based on the square you rewrite the parenthesis and combine like terms.
Factored to standard: You combine the like terms using the area diagram. For example, you would take the x+2 and put it on one side of the diagram, and do the same for x+4. You multiple the numbers and get 8. You then combine the x's and add 2+4, and 8 which we got from multiplying 2x4.
Standard to factored: To convert standard to factored, you reverse and work backwards. You know that if p and q are added they equal the b constant. They have to be multiplied by values that equal to the a constant. You factor the a constant out. You would have to guess and check for the values.
For Vertex to standard you have to take the (x+3)^2 and do (x+3) two times next to each other because the ^2 tells us so. You bring down the -2, and , you multiply 3 times 3 and get 9. You combine the like terms to get (x^2 +6x+9)-2. You are left with two x's which you can write down as y=x^2+6x+7
Standard to vertex: To turn an equation from standard to vertex you take a out and group the ax^2 and bx together. By using the area diagram you fill out the missing numbers and subtract the added term. You take the subtracted term and multiply by a. Based on the square you rewrite the parenthesis and combine like terms.
Factored to standard: You combine the like terms using the area diagram. For example, you would take the x+2 and put it on one side of the diagram, and do the same for x+4. You multiple the numbers and get 8. You then combine the x's and add 2+4, and 8 which we got from multiplying 2x4.
Standard to factored: To convert standard to factored, you reverse and work backwards. You know that if p and q are added they equal the b constant. They have to be multiplied by values that equal to the a constant. You factor the a constant out. You would have to guess and check for the values.
Why area diagrams are important:
Creating area diagrams help you visualize the equations, and you are able to understand the problems better. Completing the squares help make the steps easier and clearer.
Creating area diagrams help you visualize the equations, and you are able to understand the problems better. Completing the squares help make the steps easier and clearer.
Solving Problems with Quadratic Equations
There are three types of problems that we can solve by using quadratic equations: Kinematics, Geometry, and Economics,
Kinematics: An example of a kinematics problem we did is the rocket launch problem in the beginning of this project. We started with the equation h(t)=-16t^2+92t+160. We simplified the equation by trying to find the coordinates of the vertex and the positive x intercept. If we convert into vertex form to get the vertex and we get the factored form for the x intercept we can complete the equation. h(t)=-16(t-2.875)^2+292.25.
The maximum height of the rocket is 292.25, the amount of passed time until maximum height is 2.875, and the x intercept is 7.14883
An example of a geometry problem is the Leslie's flowers worksheet where we had to help a gardener plan the layout of her garden box, because she had use triangular flowerbeds. For economics the worksheet called Widgets asked us to solve for predicated sales of a company and we had to solve for the gained profit, and their price was "x".
There are three types of problems that we can solve by using quadratic equations: Kinematics, Geometry, and Economics,
Kinematics: An example of a kinematics problem we did is the rocket launch problem in the beginning of this project. We started with the equation h(t)=-16t^2+92t+160. We simplified the equation by trying to find the coordinates of the vertex and the positive x intercept. If we convert into vertex form to get the vertex and we get the factored form for the x intercept we can complete the equation. h(t)=-16(t-2.875)^2+292.25.
The maximum height of the rocket is 292.25, the amount of passed time until maximum height is 2.875, and the x intercept is 7.14883
An example of a geometry problem is the Leslie's flowers worksheet where we had to help a gardener plan the layout of her garden box, because she had use triangular flowerbeds. For economics the worksheet called Widgets asked us to solve for predicated sales of a company and we had to solve for the gained profit, and their price was "x".
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A problem I have done in this unit is called "Is it a Homer?" which was a worksheet we completed. This problem tells us that when a baseball is thrown or hit, its air path is almost a perfect parabola. When Mighty Casey hits it, the ball reaches its maximum height of 80 feet and is 200 feet away from the home plate. The center field fence is 380 feet from the home plate and is 15 feet tall. The problem asks us if casey's hit clear the fence for a home run. First we start with y=a(x-h)^2+k. We plug in 200 for h and 80 for k, which turns the equation into y=a(x-200)^2+80. Then we fill in 0 for x and y, which gives us 0=a(0-200)^2+80. You subtract 80 from both sides, and we have -80=a(-200)^2 so we divide by -200^2. Our answer would be y=500(x-200)^2+80 after we have pit it back into vertex form. So the answer to the problem is that it does clear.
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Reflection/Habits of a MATHEMATICIAN:
It would be believed that I would have comprehended the forms, functions, and notations of quadratic expressions, I still do not fully understand how to calculate nor execute the transitions between vertex, standard, and factored forms of the equations. This math unit was not only rushed, but not in depthly explained. I feel as though we got worksheets on top of worksheets with multiple due dates and deadlines to have things completed that some were unable to actually understand what was being taught. Some may say to just ask questions, but due to incidents in class, I feel like I’ve shied away from asking, because other students tease. There are always things I can do to get help and I will admit I didn't take full advantage of that. In the upcoming Junior year, I am very nervous as I feel I am not ready for the concepts to come. The backbone of the 11th grade math skills come from Sophomore year. I can't say that I've been taught everything or understood what was attempted to be taught. I’m very nervous about college academics and the ACT. For the upcoming year, I hope that we can focus more on the ACT as a whole, as opposed to individual concepts for long chunks of time. In conclusion, for my Sophomore year math, I think it could have been taught with more explanation that was applicable all students learning styles.
Habits of a MathematicianLooking for patterns- I started seeing similarities between linear functions and quadratic functions with some of the equations like
y=mx+b and y=ax^2+bx+c
Starting small- There were a lot of confusing concepts in this unit, so I knew that if I were to take it one problem at a time, I could get it done faster because I wouldn't be overwhelmed with all the other problems.
Being systematic- By using the proper equations for different sections, I was able to complete the assigned tasks accurately and on time. I knew that I had to take a step back and highlight what equations would be used and where.
Taking apart and putting back together- I would keep the problems separate in order to really be able to break things down. I looked at other worksheets in order to make sure I was on the right track.
Conjecturing and testing- Every so often there would be a problem I would look at and now understand it at all. I would take the things I learned and use an estimation initially, and eventually would be able to come to an accurate conclusion.
Staying organized- I knew that I had to keep all my work clear and concise in order to be able to obtain what was written. I tried to keep all my notes broken down into categories. That would help me be able to quickly look back and find what I was looking for
Describing and articulating- I learned the importance of having a description of how you got to an answer, not just the answer itself because then if there were mistakes, it was easy to look back on the work and find where the mistake had been made and fixing it.
seeking why and proving- Showing how I got to a conclusion was really beneficial to me so I could apply that the future problems because I would just repeat most of the same steps.
Being confident, persistent, and patient- I was constantly getting frustrated with problems because I didn't understand how to solve them. It was really hard for me to be able to move from one problem to another if I felt like I didn't completely understand it. It made me feel really low. But, by taking my time with each problem and asking for help when needed, pushed me through the hard parts.
Collaborating and listening- This came to my benefit a lot because in the beginning I had a really hard time understanding how to convert all the formulas. With the help from my peers, I was able to come to a better understanding because I got to see it through a different perspective.
Generalizing- I learned that generalizing problems helped a lot because equations became applicable to other problems. I saw this especially in the standard, vertex, and factored form problems.
Habits of a MathematicianLooking for patterns- I started seeing similarities between linear functions and quadratic functions with some of the equations like
y=mx+b and y=ax^2+bx+c
Starting small- There were a lot of confusing concepts in this unit, so I knew that if I were to take it one problem at a time, I could get it done faster because I wouldn't be overwhelmed with all the other problems.
Being systematic- By using the proper equations for different sections, I was able to complete the assigned tasks accurately and on time. I knew that I had to take a step back and highlight what equations would be used and where.
Taking apart and putting back together- I would keep the problems separate in order to really be able to break things down. I looked at other worksheets in order to make sure I was on the right track.
Conjecturing and testing- Every so often there would be a problem I would look at and now understand it at all. I would take the things I learned and use an estimation initially, and eventually would be able to come to an accurate conclusion.
Staying organized- I knew that I had to keep all my work clear and concise in order to be able to obtain what was written. I tried to keep all my notes broken down into categories. That would help me be able to quickly look back and find what I was looking for
Describing and articulating- I learned the importance of having a description of how you got to an answer, not just the answer itself because then if there were mistakes, it was easy to look back on the work and find where the mistake had been made and fixing it.
seeking why and proving- Showing how I got to a conclusion was really beneficial to me so I could apply that the future problems because I would just repeat most of the same steps.
Being confident, persistent, and patient- I was constantly getting frustrated with problems because I didn't understand how to solve them. It was really hard for me to be able to move from one problem to another if I felt like I didn't completely understand it. It made me feel really low. But, by taking my time with each problem and asking for help when needed, pushed me through the hard parts.
Collaborating and listening- This came to my benefit a lot because in the beginning I had a really hard time understanding how to convert all the formulas. With the help from my peers, I was able to come to a better understanding because I got to see it through a different perspective.
Generalizing- I learned that generalizing problems helped a lot because equations became applicable to other problems. I saw this especially in the standard, vertex, and factored form problems.